Fractional Order Constrained Mechanical System Noether, Symmetry And Lie Symmetry Theory Research

In this paper, we use Lie group method to research the Noether and Lie symmetrical theories of fractional constraint dynamical systems for the first time. We established the Noether and Lie symmetrical theories of fractional nonconservative Lagrange systems, and set up the Noether and Lie symmetrical theories of fractional Hamilton systems. This paper mainly attributed to the following aspects:Firstly, we established the fractional Euler-Lagrange equations for conservative systems. In virtue of the fractional Hamilton’s principle of conservative systems, and combined with the exchanging relationship between fractional operators and isochronous variation, the fractional Euler-Lagrange equations of conservative systems are obtained. This conclusion is consist with Agrawal’s.Secondly, we established the fractional Euler-Lagrange equations for nonconservative systems. According to the fractional Hamilton’s principle of nonconservative systems, we derived the fractional Euler-Lagrange equations of these systems. In addition, we derived the fractional Hamiltonian canonical equations.Thirdly, we set up the nonholonomic systems’fractional Euler-Lagrange equations. Introducing nonholonomic constraints of Apell-Chetaev type, and using fractional d’Alembert-Lagrange principle, the fractional Euler-Lagrange equations of these systems are got.Fourthly, Noether symmetrical theories of fractional nonconservative Lagrange systems are established. As Noether symmetrical theory is based on the invariant of Hamiltonian action under the infinitesimal transformations, and combined with the variational formulae of Hamiltonian action, the definitions and criteria of fractional Noether symmetrical transformations of these systems are given. Then, the fractional Noether theorem and corresponding conserved quantity are obtained.Fifthly, Noether symmetrical theories of fractional Hamilton systems are established. As Noether symmetrical theory is based on the invariant of Hamiltonian action under the infinitesimal transformations, and in virtue of variational formulae of Hamiltonian action, the definitions and criteria of fractional Noether symmetrical transformations of these systems are given. Then, the fractional Noether theorem and corresponding conserved quantity are obtained.Sixthly, the Lie symmetrical theories of fractional nonconservative Lagrange systems are established. As the Lie symmetrical theory is based on the invariant of the differential equations of the system under the infinitesimal transformations, by introduce the differential operator of infinitesimal generators, the determining equations are got. Then giving the definition of Lie symmetrical transformations, and derived the conserved quantity of the system.Seventhly, the Lie symmetrical theories of fractional Hamilton systems are established. As the Lie symmetrical theory is based on the invariant of the differential equations of the system under the infinitesimal transformations, by introduce the differential operator of infinitesimal generators, the determining equations are got. Then giving the definition of Lie symmetrical transformations, and derived the conserved quantity of the system.Finally, we summarize the main results of our research and envision the future research directions.